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© Copyright 2001, Jim Loy
Here is a classic problem. Little Darius and little Xerxes were 10 km apart. They rode their bicycles at a constant 10 km/hr toward each other. As they started off, a fly took off from one bicycle and flew to the other bicycle at 20 km/hr, then it flew back to the other bicycle, and then back and forth until it was crushed between the two bicycles when the two bicycles collided. Pretty stupid, all around. How far did the fly fly?
Answer:
There is more than one way to solve this. First, let's look at the easy way. The two bicycles were 10 km apart. They crash halfway between them in half an hour. The fly travels 10 km in that half an hour. That's the answer.
How about the hard way? The fly goes twice as fast as the either bicycle. So the fly initially covers 20/3 km while the bicycle A covers 10/3 km. The fly turns around. Bicycle B is now 10/3 km away. The fly returns to bicycle B, going twice as fast as bicycle B. So the fly goes 20/9 km while bicycle B goes 10/9 km. Then the fly turns around and goes 20/27 km, while bicycle A goes 10/27 km. After an infinite number of flights, the fly has gone 20/3+20/9+20/27+20/81+... That appears to be a Geometric Series. The sum is S=a/(1-r), where a is the first term 20/3, and r is the ratio of successive terms 1/3. So the sum is 10 km.
It is said that when asked to do this puzzle, John von Neumann did it the hard way, in his head, in just a few seconds. He didn't guess that there was an easy way.